Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Cumulative plastic deformation :math:`P` is equal to:

:math:`P\mathrm{=}\frac{{\sigma }_{L}\mathrm{-}{\sigma }^{y}}{C}`

with: :math:`{\sigma }_{L}`: constraint to the node in question

:math:`{\sigma }^{y}`: elastic limit

:math:`C`: slope of the traction curve

The constraints are given by:

:math:`{\sigma }_{\mathit{xx}}({N}_{i})\mathrm{=}\mathrm{-}{P}_{\mathit{res}}({N}_{i})`

The plastic deformation is given by:

:math:`\mathrm{\mid }{\varepsilon }_{\mathit{xx}}^{p}({N}_{i})\mathrm{=}P({N}_{i})\mathrm{\mid }`


Benchmark results
----------------------

Uniaxial stress, plastic deformation, and cumulative plastic deformation are calculated at nodes :math:`{N}_{2}` and :math:`{N}_{3}`.

For the problem in question:


.. csv-table::

    "", ":math:`{N}_{2}` "," :math:`{N}_{3}`"
    ":math:`{\sigma }_{\mathit{xx}}` ", "0", "—300"
    ":math:`{\varepsilon }_{\mathit{xx}}` ", "0", "—6.1658 10—2"
    ":math:`{\varepsilon }_{\mathit{xx}}^{p}` ", "0", "6.1658 10—2"



Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


1. LORENTZ E., PROIX J.M., VAUTIER I., VOLDOIRE F., F., WAECKEL F.: Introduction to thermo-plasticity in the Aster code. Course Reference Manual. EDF - DER, SCE IMA, Dept. Mechanics and Numerical Models, HI-74/96/013/0